How to Simplify a Fraction to Lowest Terms (GCD Method, Examples)
Reduce any fraction to its simplest form by dividing by the greatest common divisor. Worked examples, a quick check, negative and improper fractions.
The short answer
To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator, then divide both by it. For 84/126 the GCD is 42, so 84 ÷ 42 = 2 and 126 ÷ 42 = 3, giving 2/3. A fraction is in lowest terms when the top and bottom have no common factor other than 1. To check your answer, enter the fraction in the Fraction Calculator, multiply it by 1/1, and the result is shown in simplest form with the GCD step and a decimal.
What "simplest form" means
A fraction like 18/24 and the fraction 3/4 are the same number: both equal 0.75. Simplifying does not change the value, it only writes it with the smallest whole numbers possible. Teachers, exam answer keys and most recipes and measurements expect the simplest form, so 18/24 is usually marked as incomplete even though it is not wrong.
The rule is simple: divide the top and the bottom by the same number. Doing the same thing to both keeps the value the same, just as multiplying both by the same number does.
Method 1: Divide by the GCD (one step)
The greatest common divisor is the largest number that divides both numbers exactly.
Example: simplify 18/24.
Listing factors works for small numbers. For larger ones, the Euclidean algorithm finds the GCD faster: divide the larger number by the smaller, keep the remainder, and repeat until the remainder is 0. The last non-zero remainder is the GCD.
Example: simplify 84/126.
The HCF and LCM Calculator finds the GCD (also called the HCF, highest common factor) of two or more numbers if you want to see it on its own.
Method 2: Divide by small primes (step by step)
If finding the GCD feels hard, keep dividing by small prime numbers that go into both, until none do.
Example: simplify 45/60.
This takes more steps but you cannot get it wrong as long as you keep going until nothing else divides both. The GCD of 45 and 60 is 15, which is 5 × 3, the two numbers you divided by.
Quick divisibility checks help:
| Divide by | When it works |
|---|---|
| 2 | Both numbers are even |
| 3 | The digits of each number add up to a multiple of 3 |
| 5 | Both numbers end in 0 or 5 |
| 10 | Both numbers end in 0 |
| 11 | For two-digit numbers, both digits are the same (22, 33, 44) |
How to simplify with the Fraction Calculator
The Fraction Calculator simplifies every answer it gives, so you can use it to simplify a single fraction:
The step-by-step solution shows the multiplication (84 × 1 = 84, 126 × 1 = 126), then Simplify: GCD(84, 126) = 42 and the result 2/3. The decimal value, 0.6667 here, is shown under the result to four decimal places, which is a handy check that the simplified fraction has the same value as the original.
The calculator takes whole-number numerators and denominators. A mixed number such as 2 1/4 needs to be converted first: 2 × 4 + 1 = 9, so enter 9/4.
Special cases
Fractions that are already simplest
17/51 looks reducible, and it is: GCD(17, 51) = 17, so 17/51 = 1/3. But 17/50 cannot be simplified, because 17 is prime and does not divide 50. If the GCD is 1, the fraction is already in lowest terms.
Improper fractions
An improper fraction has a numerator larger than its denominator. Simplify it the same way: 42/12 has GCD 6, so 42/12 = 7/2. If you need a mixed number, divide: 7 ÷ 2 = 3 remainder 1, so 7/2 = 3 1/2. Many maths teachers accept 7/2 as the simplest form; check what your course expects.
Negative fractions
Simplify the numbers and keep one minus sign: −6/8 = −3/4. Put the minus sign on the numerator (or in front of the fraction), not on the denominator. If you enter a negative denominator in the calculator, it keeps the sign there, so you may see 3/−4; that is the same value as −3/4.
Zero
0/7 simplifies to 0. A fraction with 0 as the denominator is undefined, so the calculator will not calculate it.
Why simplifying first saves work
When you multiply fractions, simplifying first keeps the numbers small. For 8/12 × 9/16, simplify 8/12 to 2/3 and 9/16 stays as it is, then 2/3 × 9/16 = 18/48 = 3/8. You can also cancel across: the 3 and the 9 share 3, and the 2 and the 16 share 2, so 2/3 × 9/16 = 1/1 × 3/8 = 3/8. The full steps for adding, subtracting, multiplying and dividing fractions are in How to Add, Subtract, Multiply and Divide Fractions.
Common mistakes
- Dividing only the top or only the bottom. That changes the value. Always divide both by the same number.
- Stopping too early. 12/18 → 6/9 is correct but not finished; 6/9 → 2/3.
- Subtracting instead of dividing. (12 − 6)/(18 − 6) = 6/12, which is a different number.
- Cancelling digits. Crossing out the 6s in 16/64 gives 1/4, which happens to be right, but crossing out the 2s in 12/24 gives 1/4 when the real answer is 1/2. Always divide by a real common factor.
- Forgetting to simplify the final answer after adding or multiplying fractions.
Simplifying is just one division done twice. Find the biggest number that divides both, divide, and check that nothing else divides both.
Frequently asked questions
How do you simplify a fraction to lowest terms?
Find the greatest common divisor of the numerator and denominator and divide both by it. For 84/126 the GCD is 42, giving 2/3.
How do I know a fraction is fully simplified?
It is fully simplified when the numerator and denominator have no common factor other than 1, which means their GCD is 1.
Can I simplify a fraction with the Fraction Calculator?
Yes. Enter the fraction, choose multiply, and enter 1/1 as the second fraction. The result is shown in lowest terms with the GCD step and a decimal check.
How do you simplify an improper fraction?
The same way: divide top and bottom by their GCD. 42/12 becomes 7/2, which you can write as the mixed number 3 1/2 if needed.
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